Generalized linear mixed models (GLMMs) have become extremely popular in recent years. The main computational problem in parameter estimation for GLMMs is that, in contrast to linear mixed models, closed analytical expressions for the likelihood are not available. To overcome this problem, several approaches have been proposed in the literature. For this study we have used one quasi-likelihood approach, penalized quasi-likelihood (PQL), and two integral approaches: Laplace and adaptive Gauss-Hermite quadrature (AGHQ) approximation. Our primary objective was to measure the performances of each estimation method. AGHQ approximation is more accurate than Laplace approximation, but slower. So the question is when Laplace approximation is adequate, versus when AGHQ approximation provides a significantly more accurate result. We have run two simulations using PQL, Laplace and AGHQ approximations with different quadrature points for varying random effect standard deviation (Ɵ) and number of replications per cluster. The performances of these three methods were measured base on the root mean square error (RMSE) and bias. Based on the simulated data, we have found that for both smaller values of Ɵ and small number of replications and for larger values of and for larger values of Ɵ and lager number of replications, the RMSE of PQL method is much higher than Laplace and AGHQ approximations. However, for intermediate values of Ɵ (random effect standard deviation) ranging from 0.63 to 3.98, regardless of number of replications per cluster, both Laplace and AGHQ approximations gave similar estimates. But when both number of replications and Ɵ became small, increasing quadrature points increases RMSE values indicating that Laplace approximation perform better than the AGHQ method. When random effect standard deviation is large, e.g. Ɵ=10, and number of replications is small the Laplace RMSE value is larger than that of AGHQ approximation. Increasing quadrature points decreases the RMSE values. This indicates that AGHQ performs better in this situation. The difference in RMSE between PQL vs Laplace and AGHQ vs Laplace is approximately 12% and 10% respectively.
In addition, we have tested the relative performance and the accuracy between two different packages of R (lme4, glmmML) and SAS (PROC GLIMMIX) based on real data. Our results suggested that all of them perform well in terms of accuracy, precision and convergence rates. In most cases, glmmML was found to be much faster than lme4 package and SAS. The only difference was found in the Contraception data where the required computational time for both R packages was exactly the same. The difference in required computational times for these two platforms decreases as the number of quadrature points increases.